[Comp-neuro] Re: Attractors, variability and noise

james bower bower at uthscsa.edu
Wed Aug 20 17:49:42 CEST 2008


Dan,

Again, I do not deny, nor would I ever deny that intuition (what-ever  
it is) plays an important role in advancing human knowledge.  I don't  
however, unlike some of my colleagues, believe that intuition is a  
function of "mind", somehow separate from the experience of the brain,  
but rather is more likely to be some manifestation of a deeper and  
more complete consideration of the information the brain has access  
to, and, of course, the structure in which that information is  
embedded is important.

Thus,  two related questions:

1) In what structure is mathematical intuition grounded?  In math it  
is in the formal description of (in some sense preexisting)  
mathematical relationships. I suggest this is quite analogous to the  
preexisting structural relationships in neurons and networks (perhaps  
even directly in fact).  It would be hard to imagine a mathematician  
or a whole field of mathematics, claiming that one could develop  
working theories of math unrelated to the actual (physical??)  
structure of mathematics.

2) Perhaps even more importantly, by what mechanism and with reference  
to what information do 'intuitive leaps' (if there are), get validated  
or refuted.  Again, the reference in mathematics is to the physical  
structure (if you will) of mathematical objects.

The reference to more than 2,000 years ago, was, of course, to  
Pythagoras -- who also according to legend picked and chose the  
relationships in math that fit with his own somewhat mystical  
intuition about how things really worked.  Pythagoras also (by lore),  
believed that simplicity and regularity were a virtue in and of  
themselves.

Finally, one has to be very careful, of course, with how scientists  
and mathematicians self describe what they do.  There is a long  
history of individuals (Newton Feynman to name two), who, for whatever  
set of reasons, have wanted others to believe that they thought  
differently about things and in particular, relied heavily on an  
uncommon deep intuition.  In mathematics as you have pointed out, it  
is perhaps easier to see, or at least discover the contribution of  
intuition and gaps in the formal proof.  In models of the brain that  
have no direct reference to the physical structure of the brain itself  
-- it is much more difficult.

Jim



On Aug 20, 2008, at 7:42 AM, Dan Goodman wrote:

> (I sent this email a little while ago but it was rejected because I  
> sent it from the wrong email address. Apologies for it being  
> slightly out of sync with the discussion.)
>
> Hi Jim,
>
> Thanks!
>
> I agree that there's a big difference, a huge difference, between how
> we should be reasoning about maths and how we should be reasoning  
> about
> the brain. Actually I suspect we'll need quite new types of reasoning
> to understand the brain, probably quite different to any sort of
> reasoning we've seen before. It seems likely that computer simulation
> (at many different levels of detail) will be involved in that. But
> let's not abandon the successful examples of reasoning we already know
> about too soon - and intuitive, simplified concepts play a big role in
> those.
>
> "In this sense isn't it fair to say that what mathematicians
> fundamentally do is build the equivalent of realistic models in
> biology?"
>
> That's an interesting analogy (!), I see what you're getting at and
> it's a useful way to look at mathematics, but it's not the whole  
> story.
> Apologies for the following slightly long digression about maths. In a
> modern mathematical proof, we don't give all the steps in the  
> reasoning
> because it's practically impossible. As a personal example, in a paper
> I wrote there are two equations with "from this it follows that" in
> between them - to actually do that calculation took 12 pages of
> algebra. I believe this is pretty standard in mathematical papers, and
> in fact many papers would require significantly more work to fill in
> the steps than that. Recently, the Mizar project found a complete  
> proof
> of the Jordan Curve Theorem (a very simple mathematical theorem that
> basically says if you draw a wiggly circle that doesn't intersect
> itself it will have an inside and outside, i.e. the sort of thing
> normal people would just assume without proof), this was the result of
> 14 years of in-depth work involving lots of people and computers
> working together. The final proof was 200,000 lines of mathematical
> reasoning. That was to prove that wiggly circles have insides and
> outsides! So obviously, proofs of serious mathematical results cannot
> possibly give all the steps because nobody would read them. Instead,
> mathematics relies on the judgement and training of mathematicians -
> they are trained to know which statements can be made 'safely' and
> which can't. The idea is that 'in principle' a mathematician could
> write out the complete proof, but it's almost never done, and the
> Jordan Curve Theorem example shows just how difficult that is. Real
> mathematics absolutely relies on intuitive concepts, it would be
> impossible without them.
>
> But there's more even than that, because sometimes mathematicians rely
> on intuitions that are actually wrong! In the "golden age" of
> mathematics (in the 18th century), the great mathematician Leonhard
> Euler proved extraordinary results using infinite series. He's the one
> responsible for the infinite series for pi, that pi^2/6 is the sum of
> the reciprocals of the squares of the integers 1+1/4+1/9+... The
> trouble was that in order to "prove" this result, and many others like
> it, he had to use series that diverged as intermediate steps. It's
> known that he was aware of this problem - so we can't just say he made
> a mistake - but that his intuition told him that it was somehow
> meaningful anyway. It turns out he's right, one can take his ideas and
> make them rigorous by developing and overlaying new concepts on them.
> His 'understanding' of numbers and infinite series went beyond his
> ability to 'realistically model' them, so to speak. If you're
> interested in this you can read more about it in Morris Kline's
> excellent "Mathematical thought from ancient to modern times", volume
> 2, chapter 20, section 7, page 460 on the version that is free to read
> on Google Book Search.
>
> In modern times, there are more examples. The contemporary
> mathematician William Thurston is partly famous for his  
> "Geometrisation
> Conjecture". Thurston didn't manage to prove this theorem himself (it
> was just recently proved by Russian Mathematician Grigori Perelman
> making him eligible for the $1m Clay Mathematics Institute prize), but
> his enormous insights into the nature of three-dimensional geometry
> meant that this conjecture essentially determined the course of most  
> of
> the work in 3D geometry since 1982, when he made the conjecture, until
> now. All of his insights have proved correct, and have been invaluable
> in the development of that subject (which was my field).
>
> I'm not terribly knowledgeable about theoretical physics, but I'm told
> there are people in that subject who are in a similar position: they
> have incredible insights and intuitions about their work that go  
> beyond
> our ability to make them formal (yet!) but that they are hugely
> successful in practice. I believe Edward Witten is the person I have  
> in
> mind. Or for that matter, there's no mathematically consistent theory
> of Feynmann's path integrals, but they too are incredibly useful.
>
> So in conclusion...
>
> "Didn't they give up on the kind of "gut" intuition about the meaning
> of math sometime around two thousand years ago?"
>
> No! :-) (Although they'd like you to believe that, for some reason.)
>
> In regards to your second point, you're absolutely right that these
> intuitions don't come from nowhere, and that intuition about intuition
> guiding our intuition is a dangerous place to be in. But I don't think
> anyone is arguing that mathematicians and physicists working in the
> field should be ignorant about this stuff are they? Certainly for my
> part I feel there's an awful lot more biology I need to know.
>
> On your third point, I agree that in both physics and maths often the
> elegant theories came from an attempt to make elegant a mess of  
> complex
> and seemingly inconsistent results. But at some point, hopefully we'll
> find that elegant theory, and who's to say we're not yet in a position
> to do that? I would imagine that when this insight comes, if it does,
> the nature of it will be a big surprise (if not, we would already know
> it). The point is, we shouldn't rule out one way or another because we
> don't yet know which way will be the successful one. (Is there a
> history of unsuccessful ideas in physics and maths? There should be. I
> know that Popper wrote some interesting stuff about pre-Socratic
> science in "The World of Parmenides" but apart from a few cases of
> really well known things like the 'phlogiston theory', I don't know of
> anything more recent that talks about this.)
>
> I'll just add that I'd imagine you agree with that, but that it's  
> still
> important to make this point explicit because opinions about where the
> answers are likely to be found affect where the funding goes, and this
> is pretty important.
>
> Dan
>
>
>
> jim bower wrote:
>
> > Dan
>
> >
>
> > Great post, and I probably should have been a bit more careful in
> my wording. but a couple of things to say.
>
> >
>
> > First, human intuition when applied to mathematics is quite a
> different thing than human intuition applied to the system (the brain)
> that generates human intuition. The discussion of free will and our
> deep belief in the primacy of individual-based learning being an
> example. Intuition in mathematics is generated in the context of a
> formal description related to discovered mathematical relationships
> themselves pushing you guys into rather bizare musings . In this sense
> isn't it fair to say that what mathematicians fundamentally do is  
> build
> the equivalent of realistic models in biology? Didn't they give up on
> the kind of "gut" intuition about the meaning of math sometime around
> two thousand years ago?
>
> >
>
> >
>
> > Second, I would be the very last to suggest that intuition is not
> a key element in human progress. One reason I have expected every
> physicist entering my laboratory for years to do experiments or at
> least build realistic models is to give them a basis in biology for
> their intuitions - the question is- what is the characteristic of the
> tools that guide that intuition. If it is intuition about intuiton  
> that
> is guiding intuition, everything becomes rather circular.
>
> >
>
> > Third, while I belive it will be at least very hard, and for sure
> require new tools, I have no objection in principle to theories that
> are easier to understand. I am happy to hope this will be the case for
> biology as well (although I am skeptical for all the reasons mentioned
> previously). But the critical question for me is where those theories
> came from. Physics and math are full of examples where something the
> data pushes to be very complex and hard to understand is replaced or
> captured in something more eligent and even intuitive. . It was the
> complexty of the earth centric models that lead compernicus to propose
> a simpler (and widely less accurate) sun centered model, coupled  
> with a
> necessary fudge in the earth centric models (the equint) which  
> violated
> intuition. The question is what you do first and what you do second.  
> It
> is my contention that the chances that more abstract top down models
> will discover brain function, are much less than the possibility that
> detailed realistic models will provide the kind of brain structure
> directed intuition that could lead to some more formal elegent or
> understandable general principle. In almost all cases now (although
> there are exceptions) , more abstract modelers base their assumptions
> on descriptions of the nervous system often less sophisticated than
> those found in our textbooks (and they are very much lacking). In my
> view, it is likely to be more useful to start with a big complex
> realistic model (there are a bunch of them in model-DB and we are  
> happy
> to give you any of ours you want) and work from there.
>
> >
>
> > This is what Sharon Crook, Bard and I were starting to do 15 years
> ago in the paper I referenced previously.
>
> >
>
> > But what I was railing against was a field that holds as its
> highest achievment a one and a half page publication in Science or
> Nature that makes a clear profound (and "understandable") point, and
> whose leading journal (journal of neuroscience) limits all  
> publications
> regardless of the subject matter to 1500 words (and you can see my
> personal problem with that. ;-). The drive for short format, easu to
> understand, sound-byte science just seems so completely out of step
> with the structure we are trying to figure out.
>
> >
>
> > Jim bower
>
> >
>
> >
>
> > Sent via BlackBerry by AT&T
>
> _______________________________________________
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> http://www.neuroinf.org/mailman/listinfo/comp-neuro




==================================

Dr. James M. Bower Ph.D.

Professor of Computational Neuroscience

Research Imaging Center
University of Texas Health Science Center -
-  San Antonio
8403 Floyd Curl Drive
San Antonio Texas  78284-6240

Main Number:  210- 567-8100
Fax: 210 567-8152
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